A prediction problem in $L^2 (w)$

For a nonnegative integrable weight function $w$ on the unit circle $T$, we provide an expression for $p=2$, in terms of the series coefficients of the outer function of $w$, for the weighted $L^p$ distance $\inf_f \int_T\vert 1-f\vert^p wd \mu$, where $\mu$ is the normalized Lebesgue measure and $f$ ranges over trigonometric polynomials with frequencies in $[\{\dots,-3,-2,-1\}\setminus\{-n\}]\cup\{m\}$, $m \geq 0$, $n \geq 2$. The problem is open for $p \neq 2$.